Calculus Examples

$4 x^{2} + 9 y^{2} = 36$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$4 x^{2} + 9 {(0)}^{2} = 36$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $4 x^{2} + 9 {(0)}^{2}$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Raising $0$ to any positive power yields $0$ .

$4 x^{2} + 9 \cdot 0 = 36$

Step 1.2.1.1.2

Multiply $9$ by $0$ .

$4 x^{2} + 0 = 36$

Step 1.2.1.2

Add $4 x^{2}$ and $0$ .

$4 x^{2} = 36$

Step 1.2.2

Divide each term in $4 x^{2} = 36$ by $4$ and simplify.

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Step 1.2.2.1

Divide each term in $4 x^{2} = 36$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{36}{4}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{36}{4}$

Step 1.2.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{36}{4}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $36$ by $4$ .

$x^{2} = 9$

Step 1.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{9}$

Step 1.2.4

Simplify $\pm \sqrt{9}$ .

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Step 1.2.4.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

Step 1.2.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 3$

Step 1.2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 3$

Step 1.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3$

Step 1.2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3, - 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(3, 0), (- 3, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$4 {(0)}^{2} + 9 y^{2} = 36$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify $4 {(0)}^{2} + 9 y^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Raising $0$ to any positive power yields $0$ .

$4 \cdot 0 + 9 y^{2} = 36$

Step 2.2.1.1.2

Multiply $4$ by $0$ .

$0 + 9 y^{2} = 36$

Step 2.2.1.2

Add $0$ and $9 y^{2}$ .

$9 y^{2} = 36$

Step 2.2.2

Divide each term in $9 y^{2} = 36$ by $9$ and simplify.

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Step 2.2.2.1

Divide each term in $9 y^{2} = 36$ by $9$ .

$\frac{9 y^{2}}{9} = \frac{36}{9}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $9$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{9 y^{2}}{9} = \frac{36}{9}$

Step 2.2.2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{36}{9}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $36$ by $9$ .

$y^{2} = 4$

Step 2.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{4}$

Step 2.2.4

Simplify $\pm \sqrt{4}$ .

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Step 2.2.4.1

Rewrite $4$ as $2^{2}$ .

$y = \pm \sqrt{2^{2}}$

Step 2.2.4.2

Pull terms out from under the radical, assuming positive real numbers.

$y = \pm 2$

Step 2.2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = 2$

Step 2.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - 2$

Step 2.2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = 2, - 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 2), (0, - 2)$

Step 3

List the intersections.

x-intercept(s): $(3, 0), (- 3, 0)$

y-intercept(s): $(0, 2), (0, - 2)$

Step 4